True hard spheres do not exist. Colloidal hard spheres approximate the behaviour in many ways
Idealised hard spheres are impenetrable objects.
They are characterised by the simplest of the potentials.
For a sphere of diameter \sigma, enormous simplification of the Boltzmann factor:
e^{-\beta U(r)} = \begin{cases} 0 & \text{if } r < \sigma \\ 1 & \text{if } r \geq \sigma \end{cases}
This simplification arises because the potential ( U(r) ) is either infinite (forbidden region) or zero (allowed region).
For this reason, they hard spheres are entropy driven.
Constructing partition functions is then just counting counting valid configurations, linking geometry to thermodynamics.
Valid configurations are sphere packings:
\phi = \frac{\pi}{6} \rho \sigma^3
where \rho=N/V is the number density.
Since we have a single control parameter, the pahse diagram is one-dimensional.
Changing temeprature is immaterial to the free energy: it is a simple rescaling factor.
It is intuitive to guess the low density limit of hard spheres:
It is harder to guess the high density limit
We will gradually build our understanding from low to dense packing.
Hard spheres have a single control parameter, the packing fraction.
When treating the Asakura-Osawa depletion we introduced the excluded volume. It is also also key for hard spheres.
The distance of closest approach between two identical spheres is \sigma, which corresponds to the diameter of the spheres. The excluded volume of one particle is v_{\rm ex} =4\pi\sigma^3/3
At low densities, hard spheres are isolated and so no overlaps occcur and the accesible volume for N spheres is V_{\rm accessible} = V-Nv_{\rm ex}
For phase behaviour, we need the thermodyanamic potential. At constant volume and number of particles, the parition function is a measure of the accessible volume \mathcal{Z} = \frac{1}{N! \Lambda^{3N}} \int_{V_{\rm accessible}} d\mathbf{r}_1 \ldots d\mathbf{r}_N
where \Lambda is the thermal de Broglie wavelength \Lambda = h/\sqrt{2\pi mk_B T}.
Mass and temperature are factorising out.
We integrate over valid (non-overlapping) configurations such that |\mathbf{r}_i - \mathbf{r}_j| \geq \sigma for all i \neq j. This yields \mathcal{Z} = \dfrac{(V-Nv_{\rm ex}/2)^N}{N!\Lambda^{3N}},
with the 1/2 factor coming from the fact that we avoid double counting the excluded volume of pairs.
We immediately obtain the entropy as S = k_B \ln \mathcal{Z} = k_B \left[ N \ln(V - N v_{\rm ex}/2) - \ln N! - 3N \ln \Lambda \right]
Via Stirling approximation \ln N! = N\ln N -N we get S = k_B \left[ N \ln(V - N v_{\rm ex}/2) - (N \ln N - N) - 3N \ln \Lambda \right]
This can be rewritten as S = N k_B \left[ \ln\left( \frac{V - N v_{\rm ex}/2}{N \Lambda^3} \right) + 1 \right]
The equation of state links the three relevant thermodynamics variables P,T and \phi
P = -\left(\dfrac{\partial F}{\partial V}\right)_{N,T} =T\left(\dfrac{\partial S}{\partial V}\right)_{N,T}= \dfrac{k_B T }{v-v_{\rm ex}/2}
This expression can be simplified (do it as an exercise) to obtain the equation of state
Z_{\rm comp}= \dfrac{PV}{Nk_BT}=\dfrac{1}{1-4\phi} \quad (\phi\ll 1)
where Z_{\rm comp} is compressibility factor (to not be confused with the partition function).
For small \phi we have
Z_{\rm comp} =\dfrac{PV}{Nk_BT} = 1+4\phi+O(\phi^2)
Note
Inspect this term and recognise that this illustrates that the dilute limit is an ideal gas + a correction.
The expression Z_{\rm comp} = 1+4\phi+O(\phi^2) is just the simplest form of the more generic virial expansion, the perturbative series
Z_{\rm comp}= 1 + B_2 \rho + B_3 \rho^2+ \dots
The B_2, B_3, \dots are known as virial coeffcients and for non-hard-sphere systems they also depend on temperature, B_2(T), B_3(T),\dots
They encode correlations (pairs, triplets and so on). In a genric setting
Z_N=\frac{1}{N!\Lambda^{3 N}} \int \cdots \int \exp \left[-\beta \sum_{i<j} U\left(r_{i j}\right)\right] d \mathbf{r}_1 \ldots d \mathbf{r}_N
can be re-written using the Mayer function f_{i j}=e^{-\beta u\left(r_{i j}\right)}-1 and e^{-\beta \sum_{i<j} U\left(r_{i j}\right)}=\prod_{i<j}\left(1+f_{i j}\right) yielding
Z_N=\frac{1}{N!\Lambda^{3 N}} \int \cdots \int \prod_{i<j}\left(1+f_{i j}\right) d \mathbf{r}_1 \ldots d \mathbf{r}_N
One can expand the product
\prod_{i<j}\left(1+f_{i j}\right)=1+\sum_{i<j} f_{i j}+\sum_{i<j, k<l}^{\text {distinct }} f_{i j} f_{k l}+\ldots The first term is the ideal gas, the second is clearly pair correlations (only pairwise distances).
The virial expansion can also be represented diagrammatically. For the compressibility factor ( Z_{} ), we have:
This motivates one to define B_2(T)=-\frac{1}{2} \int f(r) d \mathbf{r}=-2 \pi \int_0^{\infty}\left[e^{-\beta U(r)}-1\right] r^2 d r
For hard spheres this results in B_2 = \frac{2\pi}{3} \sigma^3
So Z = 1 + \dfrac{2\pi}{3} \sigma^3\rho = 1+ 4\left( \dfrac{\pi}{6} \sigma^3\rho\right)=1+4\phi as we calculated earlier.
Important
We can simply focus on the accessible volume and ignore overlap between exclusion volumes
We can focus on the configurational entropy and extract the equation of state relating P,T,\phi
The compressibility factor in terms of the packing fraction \phi is the most condensed expression and is Z_{\mathrm{comp}}=\frac{P V}{N k_B T}=1+4 \phi+O\left(\phi^2\right) highlighting the first non-trivial perturbation to the ideal gass
We can see this as an instance of the more general virial and cluster expansion formalism. which can used for any fluid. Z_{\text {comp }}=1+B_2 \rho+B_3 \rho^2+\ldots
The first nontrivial correction is the second virial coefficient B_2(T)=-2 \pi \int_0^{\infty}\left[e^{-\beta U(r)}-1\right] r^2 d r
As we increase the packing fraction, the accessible (free) volume reduces rapidly and tehrmal motion is hindered.
Tight disordered (random) packing of spheres are described as jammed: link with glasses (in future lectures).
Densest packing reaches a maximum packing of around \phi_{\mathrm{rcp}} \approx 0.64: this is not unique and depends on the protocol of preparation.
Kepler’s conjecture (1611, proved only in 2017):
Densest packings are Face Centerd Cubic (FCC) or Hexagonally Cosed Packed (HCP) \phi_{\max }=\frac{\pi}{3 \sqrt{2}} \approx 0.74
YES
NO
Early computer simulations ( molecular dynamics, Alder and Wainwright 1957, Monte Carlo, Wood and Jacobson (1957)) proved that that crystalisation is possible.